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Chapter 1: Problem 44

Sketch the given set on a number line. $$\\{x | 2

Short Answer

Expert verified
Shade the number line between 2 and 7, using open circles at both points.

Step by step solution

01

Understand the Inequality

The given set is \( \{x | 2<x<7\} \). This means we are interested in all the real numbers between 2 and 7, but not including 2 and 7 themselves.
02

Draw a Number Line

Draw a horizontal line and mark the points 2 and 7 on this line. These points will help in identifying the range of values we need to consider.
03

Indicate the Range on the Number Line

Use open circles to indicate that the endpoints 2 and 7 are not included in the set. An open circle at a point means the point itself is not part of the set but points close to it are.
04

Shade the Region Between the Points

Shade the region on the number line between the open circles at 2 and 7 to show that all numbers between these two points are included in the set.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Number line
A number line is a visual representation of numbers placed along a straight line. The line helps us to easily understand the size and order of the numbers. When we sketch on a number line, we draw a horizontal line and mark important points, or numbers, along this line. In our exercise, we are interested in the range between 2 and 7. We place these points on the line to help visualize our inequality.
  • Draw a straight, horizontal line.
  • Mark the points 2 and 7 on the line.
  • This helps us to know where we will focus our attention.
Inequality notation
Inequality notation tells us about the relationship between two values. In the given exercise, we have the set \( \{x | 2 < x < 7\} \). This notation shows that you are looking for all numbers \( x \) that are greater than 2 and less than 7, but not including 2 and 7 themselves. Here is how you can break it down:
  • The symbol \( < \) means 'less than'.
  • The symbol \( > \) means 'greater than'.
  • This inequality does not include the endpoints鈥�2 and 7.
So, you need to show all values between 2 and 7, excluding these bounds.
Open circles
Open circles are used on a number line to indicate that a number is not included in a set. In our inequality \( 2 < x < 7 \), we need to show that 2 and 7 are not part of our solution, even though they boundary the range. Here is why and how we use open circles:
  • Draw an open circle on the number line at 2.
  • Draw another open circle at 7.
  • An open circle means the point itself is not included.
This way, anyone looking at the number line can see that 2 and 7 are the limits but are not part of the solution.
Shading regions
Shading regions on a number line helps to visually represent where the solutions to an inequality lie. Once you have marked your open circles at 2 and 7, you need to show all numbers between them. Here are the steps to shade the region:
  • Start from an open circle at 2 and shade towards the right.
  • Continue shading until you reach the open circle at 7.
  • Make sure the shaded region lies only between the open circles, not beyond.
This shaded part represents all values that satisfy \( 2 < x < 7 \). It visually includes every number between 2 and 7, excluding 2 and 7 themselves.

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Most popular questions from this chapter

Compute the given expression. Round off your answer to two decimal places where necessary. $$12.8-9.2(3.4-1.6)$$

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